Method and apparatus for chemical analysis of fluids

ABSTRACT

An apparatus and method for electrochemical fluid analysis comprises a chamber ( 1202 ) having a depth dimension for accommodating a volume of a fluid under test, first and second electrodes (A1) disposed within the chamber and extending along the depth dimension in spaced relation with each other, and a soluble solid, such as an annealed polymer, e.g. EUDRAGIT occupying a lateral gap between the first and second electrodes. The rate of dissolution as monitored by electrochemical impedance spectroscopy (EIS) of the soluble solid within the fluid depends on the chemical concentration of a corresponding analyte present in solution in the fluid. In one embodiment a silicon-based integrated circuit device defining an upper margin includes an array of electrodes disposed along said upper margin to permit direct exposure of the electrode array to the fluid under test. The device is constructed using CMOS technology.

FIELD OF THE INVENTION

The present invention relates to a system, device and method formeasuring the concentration of chemical components present within afluid and, more particularly, to systems, devices and methods formeasuring analyte concentration electrochemically.

BACKGROUND OF THE INVENTION

Chemical concentration of an analyte in a fluid can be measured bytransducing presence of the analyte into measurable physical parameters.For example, the concentration of an analyte solution can be determinedvia such techniques as spectroscopy, chromatography, calorimetry, oroptical fluorescence.

Further concentration measurement techniques involve probing theelectrical characteristics of the analyte solution. Some such techniquesinvolve coultometry. Others involve amperometric, voltametric, and/orpotentiametric titration. Many such techniques are capable of a highdegree of accuracy, speed (e.g., throughput), and efficiency.Unfortunately, the equipment required to implement such techniques cantend to be both large and bulky. As a result, the use of such equipmentis typically limited to a laboratory setting, and technicians in thefield who seek to make concentration determinations via measurement ofelectrical characteristics are often left with few attractive options.

Despite efforts to date, a need remains for effective methods andsystems for electrochemical measurement of analyte concentration,particularly for applications in which a high premium is placed onportability and field use availability. These and other needs aresatisfied by the methods and systems disclosed herein.

SUMMARY OF THE INVENTION

Apparatus and methods for electrochemical analysis of fluids areprovided according to the present disclosure.

In an exemplary embodiment of the present disclosure, an apparatus isprovided that includes a chamber having a depth dimension foraccommodating a volume of a fluid under test, a first electrode disposedwithin the chamber and extending therewithin along the depth dimension,a second electrode disposed within the chamber and extending therewithinalong the depth dimension in laterally spaced relation with the firstelectrode, and a soluble solid disposed within the chamber between thefirst and second electrodes so as to substantially completely occupy alateral gap therebetween to an extent of at least a portion of the depthdimension. A rate of dissolution of the soluble solid within the fluidis at least partially dependent on a chemical concentration of acorresponding analyte present in solution in the fluid.

A method for electrochemical analysis of fluids is also provided. Inexemplary embodiments of the present disclosure, the method includesexposing a soluble solid to a fluid, measuring a rate of dissolution ofthe soluble solid in the fluid, and determining a chemical concentrationof a corresponding analyte present in solution in the fluid based on themeasured rate of dissolution.

Additional advantageous features, functions and applications of thedisclosed apparatus and methods for electrochemical analysis of fluidswill be apparent from the description which follows, particularly whenread in conjunction with the appended figures.

BRIEF DESCRIPTION OF THE DRAWINGS

To assist those of skill in the art in making and using the disclosedsystems and methods, reference is made to the accompanying figures,wherein:

FIG. 1 is a schematic representation of an embodiment of an analyteconcentration measurement tool in accordance with the presentdisclosure;

FIG. 2 is a downward perspective view of a CMOS die useable to fabricatethe FIG. 1 measurement tool in accordance with the present disclosure;

FIG. 3 is a top plan view of the FIG. 2 CMOS die after modification viaformation on an upper margin thereof of a metallic contact pattern inaccordance with the present disclosure;

FIG. 4 is a section view of the FIG. 3 modified CMOS die taken alongsection line 4-4 shown in FIG. 3;

FIG. 5 is a downward perspective view of the FIG. 3 modified CMOS die;

FIG. 6 is a top plan view of the FIG. 3 modified CMOS die after furthermodification via formation atop the metallic contact pattern thereof ofan array of paired electrodes in accordance with the present disclosure;

FIG. 7 is a section view of the FIG. 6 modified CMOS die taken alongsection line 7-7 shown in FIG. 6;

FIG. 8 is a downward perspective view of the FIG. 6 modified CMOS die;

FIG. 9 is a top plan view of the FIG. 6 modified CMOS die after furthermodification via formation atop the array of paired electrodes thereofof a dielectric material layer in accordance with the presentdisclosure;

FIG. 10 is a section view of the FIG. 9 modified CMOS die taken alongsection line 10-10 shown in FIG. 9;

FIG. 11 is a downward perspective view of the FIG. 9 modified CMOS die;

FIG. 12 is a top plan view of the FIG. 9 modified CMOS die after fillingof the cylindrical chambers thereof with polymeric materials andassociated annealing to form an embodiment of the analyte concentrationmeasurement tool of FIG. 1 in accordance with the present disclosure;

FIG. 13 is a section view of the FIG. 12 analyte concentrationmeasurement tool;

FIG. 14 is a section view of a fluid-polymer filled cylinder of the FIG.12 analyte concentration measurement tool in accordance with the presentdisclosure;

FIG. 15 is a section view of a fluid filled cylinder of the FIG. 12analyte concentration measurement tool in accordance with the presentdisclosure;

FIG. 16 is a schematic diagram of an exemplary electrical circuitcorresponding to the fluid-polymer filled cylinder of FIG. 14; and

FIG. 17 is a schematic diagram of an exemplary electrical circuitcorresponding to the fluid-filled cylinder of FIG. 15.

FIG. 18 is a schematic diagram of an exemplary electrical circuit.

DETAILED DESCRIPTION OF THE INVENTION

An apparatus for electrochemical analysis of fluids is provided that canbe adapted to be compact in size, economic to manufacture, andconvenient to deploy. Exemplary embodiments of apparatus forelectrochemical analysis of fluids include a chamber having a depthdimension for accommodating a volume of a fluid under test, and a pairof electrodes disposed within the chamber and extending along the depthdimension thereof in laterally spaced relation to each other. A solublesolid is disposed within the chamber between the electrodes, occupying alateral gap therebetween to an extent of at least a portion of the depthdimension of the chamber. A rate of dissolution of the soluble solidwithin the fluid is at least partially dependent on a chemicalconcentration of a corresponding analyte present in solution in thefluid To the extent that the soluble solid dissolves in the fluid, thefluid fills the void generated by the dissolving solid. Because thesoluble solid is a poorer conductor compared to the fluid, dissolutionof the soluble solid leads to an increase of conductance between theelectrodes. The rate of conductance change further depends on theproperties of the dissolving solid and the actual analyte concentrationin solution in the fluid.

Materials suitable for use with respect to the soluble solid accordingto the present disclosure include commercially available materials thatexhibit respective solubilities dependent on the concentration insolution of a chemical component or active species of interest, e.g., H+concentration (i.e., pH), proteins, amino acids, glucose, enzymes andother analytes of interest. Exemplary materials for use with respect tothe soluble solid according to the present disclosure include polymersthat exhibit a pH-dependent dissolution rate, such as EUDRAGIT acrylicpolymers manufactured by Degussa GmbH, and polymers that exhibitdissolution rates that are dependent on the presence of colon enzyme,such as azo polymers used by Alizyme plc (Cambridge, United Kingdom).

Apparatus and methods for electrochemical analysis of fluids inaccordance with the present disclosure may be used to measure theconcentration for a large number of chemical components present within afluid under test. In embodiments of the present disclosure, suchapparatus and methods rely on polymers with specific solubilitydepending on concentration of compounds mixed within the fluid, andinclude an electronic device that allows an accurate measurement of thesolubility based upon complex conductance measurements. The lifetime ofthe electronic device may be limited in accordance with embodiments ofthe present disclosure, and controlled by processing parameters of thedevice.

In accordance with some embodiments of the present disclosure, a small,simple, energy efficient ‘lab-on-a-chip’ solution is provided having aresponse time in the field at least comparable to, if not superior to,many larger, more bulky systems commonly limited to use within alaboratory. Such an apparatus can be implemented through the use of anintegrated circuit (IC) electronic device combined with an array ofconfined micro-cylinders fabricated via MEMS processes at the surface ofa die associated with the IC electronic device, and filled with polymershaving known etching rate versus chemical concentration of activespecies in solution in the fluid under test.

For illustration purposes, the disclosed apparatus and methods aredescribed in greater detail herein with reference to a tool formeasuring analyte concentration in solution in a fluid under test.However, the disclosed systems and methods have wide rangingapplicability, as will be readily apparent to persons skilled in theart, including implementations directed to a variety of analytes. Thus,in one exemplary embodiment of the present disclosure, the apparatusincludes a soluble solid in the form of a polymer that does not dissolveuntil the pH is above a threshold value and, as a result, theconductance between the electrodes does not increase unless the fluidunder test has a pH above this threshold. If the pH of the fluid undertest is above the applicable threshold, the conductance betweenelectrodes will advantageously increase proportionally to the differencebetween the actual pH value of the fluid under test and the lowerthreshold pH of the soluble polymer.

Additionally, in one exemplary embodiment of the present disclosure, theapparatus includes a soluble solid in the form of a polymer that doesnot dissolve unless the pH of the fluid under test is below a thresholdvalue and, as a result, the conductance between the electrodes does notincrease unless the fluid under test has a pH below this threshold. Ifthe pH of the fluid under test is below the applicable threshold, theconductance between electrodes will advantageously increaseproportionally to the difference between the actual pH value of thefluid under test and the higher threshold pH of the soluble polymer.Therefore, by monitoring a rate of change in conductance between theelectrodes between which the soluble solid is disposed, either or bothof a pH limit value and an actual pH value can be derived.

When an apparatus for electrochemically analyzing a fluid based on thisprinciple of concentration-dependent solubility is used, conductancebetween each pair of electrodes may be measured as a function of time,and the rate of conductance change may be used to derive theconcentration value of the analyte present in solution in the fluidunder test. One unique advantage of such an apparatus forelectrochemically analyzing a fluid is that the apparatus can beoperated without absolute calibration. Variation in manufacturingprocess and environmental conditions, such as overall conductivity ofthe fluids under test, can cause variation in absolute conductancebetween electrodes. These variations, however, do not interfere withderivation of the concentration value of an analyte present in solutionin a fluid under test because the concentration value is determined bythe change rate of conductance, not by the absolute value ofconductance. Of course, such an apparatus can be used in conjunctionwith a reference electrode to account for environmental changes in therate of conductance.

An apparatus 100 for measuring analyte concentration in solution in afluid under test in accordance with embodiments of the presentdisclosure is shown in FIG. 1. The apparatus 100 may include asilicon-based integrated circuit (IC) 102. The IC 102 may incorporate aninput/output (IO) data block 104, a data processor and control unit(DPCU) 106, an amplitude and frequency control unit (AFCU) 108, acomplex admittance measurement unit (CAMU) 110, and an electrodeselector (ES) 112. The apparatus 100 may further include an electrodearray (EA) 114.

The IO 104 may be an interface of the circuit with respect to externaldevices. The EA 114 is a matrix of electrodes present at an upper marginor surface of the IC 102. Each of the electrodes of the EA 114 can beconnected through the ES 112 block to corresponding measurement ports ofthe CAMU 110. All other electrodes of the EA 114 may be grounded. The ES112 may be an array of analog switches which allows the selection of asingle electrode out of the EA 114. The CAMU 110 can measure the complexadmittance of the circuit connected at the selected electrode from theEA 114. The frequency and amplitude of the test signal can be controlledand/or imposed by the AFCU 108. The DPCU 106 may receive analog signalsprovided by the CAMU 110 and convert the same to digital values. TheDPCU 106 may further store and/or process such digital values, takedecisions regarding the frequencies and amplitudes of operations fromthe AFCU 108 and operate the ES 112 accordingly. The DPCU 106 mayfurther be employed to transfer to the IO 104 measurement results withrespect to concentration(s) of one or more analytes present in solutionin the fluid under test.

In accordance with embodiments of the present disclosure, the CMOS die200 shown in FIG. 2 can embody the FIG. 1 IC 102. For example, the CMOSdie 200 may include an upper margin 202 featuring an array ofperipherally-disposed contacts 204 associated with an input/outputinterface of the circuit (e.g., associated with the FIG. 1 IO 104).

A MEMS process may be utilized to modify and/or convert the CMOS die 200of FIG. 2 to form an embodiment of the FIG. 1 apparatus 100 inaccordance with the present disclosure. An example of such a process isshown and described below with reference to FIGS. 3-13.

Referring now to FIGS. 3, 4 and 5, a modified CMOS die 300 can be formedby modifying the FIG. 2 CMOS die 102 via conventional metal depositionprocess and associated appropriate patterning to form a contact pattern302 on an upper margin 304 of the die 300 operative to permit electricalinterconnection between an IC (e.g., FIG. 1 IC 102) and an electrodearray (e.g., FIG. 1 EA 114) in accordance with the present invention.

Referring now to FIGS. 6, 7 and 8, a modified CMOS die 600 can be formedby further modifying the FIG. 3 modified CMOS die 300 via an appropriatealuminum-silicon deposition and etch processes (e.g., with a highlyselective RIE) to form a mask 602. The mask 602 may include an array ofchambers 604 for accommodating small volumes of a liquid under test,each of which may include a cylindrically shaped microbarrel 606connected to ground and a column shaped central electrode 608. Thecentral electrodes 608 may be disposed within the microbarrels 606, and,further may be coaxial with, and/or coextensive (e.g., depthwise)therewith.

Referring now to FIGS. 9, 10 and 11, a modified CMOS die 900 can beformed by further modifying the FIG. 6 modified CMOS die 600 via anappropriate material layer deposition and etch process, e.g., to form adielectric material layer 902 atop the FIG. 6 mask 602. For example, thedielectric material layer 902 may be a SiO₂-Si₃N₄ layer. One or more ofthe FIG. 6 chambers 604 may be masked during this step so as to preventthe dielectric material layer 902 from extending thereto. For example,four such chambers (e.g., chambers lacking a layer 902 of a dielectricmaterial) may be positioned on respective corners 904, 906, 908, 910 ofthe die 900, and/or may be used to measure air admittance (e.g., as partof a measurement control function).

Referring now to FIGS. 12 and 13, an analyte concentration measurementtool 1200 is shown. For example, the tool 1200 may be implemented toembody the analyte concentration measurement tool 100 of FIG. 1. Thetool 1200 can be formed by further modifying the FIG. 9 modified CMOSdie 900 via filling one or more of the FIG. 6 chambers 604 with similarand/or different polymers and executing an appropriate annealing processto form fluid-polymer filled cylinders or chambers 1202 (describedfurther hereinbelow), wherein the dissolution rate of each such polymermay be specific to one or more of the same or different chemicalcompounds in solution in a fluid under test. The FIG. 9 corner-disposedchambers 904, 906, 908, 910 may be left unfilled with polymer forpurposes of measuring air admittance as part of a measurement controlfunction. Likewise, an entire row 1204 of chambers 1206, also referredto herein as fluid filled cylinders or chambers 1206, may be leftunfilled with polymer for purposes of measuring an admittance of thefluid under test as described more fully below.

The tool 1200 includes an IC 1208 which can embody, for example, theFIG. 1 IC 102, and a MEMS region 1210 which can embody the FIG. 1 EA114. The MEMS region 1210 can be configured to be exposed to the fluidunder test, while the IC 1208 can be configured such that itsinternally-disposed electrical circuitry and/or functions are securedfrom damage from the fluid under test.

Referring now to FIG. 14, when exposed to fluid 1400 from the fluidunder test, the soluble solid 1402 (e.g., polymer) within the chamber1202 is dissolved by the analyte present in solution in the fluid undertest. FIG. 14 gives a vertical plane cross-section through afluid-polymer filled cylinder (FPC) 1202 after the soluble solid 1402contained therein was etched to an etch value equal to h_(fluid). FIG.15 gives a vertical plane cross-section through a fluid filled cylinder(FC) 1206. When exposed to the fluid under test, the fluid-filledcylinders 1206, containing no soluble solid (e.g., containing nopolymer), are filled with fluid 1500. The FIG. 12 tool 1200 measures thecomplex admittances for all of the FPCs 1202 and FCs 1206 of the MEMSregion 1210 at specific time intervals and estimates the electricalproperties of polymers within the FPCs 1202 and the fluid within the FCs1206 and the etch rate for every type of polymer from every FPC 1202.Based on the etching rate, one may determine the concentrations ofactive chemical species within the fluid under test. The capacity tomake such determinations may last as long as the polymers within theFPCs 1202 are not completely dissolved within the fluid under test.

Exemplary electric equivalent circuits 1600, 1700 for FPCs 1202 and FCs1206 are given, respectively, in FIGS. 16 and 17.

The below-described algorithm is executable by a processor (e.g., by theFIG. 1 DPCU 106) in accordance with the present disclosure to allow thedetermination at a time tj of the concentration of species present insolution in the fluid under test, given the FPC 1202 are filled withdifferent polymers with etching rates specific to every chemicalconcentration of the species.

V_(probe) is a voltage source V_(probe)=V₀e^(i(ωt+φ)) with i=√−1, V₀:amplitude, ω: angular frequency, φ: phase, t: time. The compleximpedeance of the circuit from FIG. 8 (seen by the voltage sourcevoltage source V_(probe) is:

$\begin{matrix}{{Z_{g} = \frac{\begin{matrix}\left( {X_{{Cfe}\_ i} + X_{{Cfe}\_ o} + \frac{X_{Cfluid}R_{fluid}}{X_{Cfluid}R_{fluid}}} \right) \\\left( {X_{{Cpe}\_ i} + X_{{Cpe}\_ o} + \frac{X_{Cpolymer}R_{polymer}}{X_{Cpolymer} + R_{polymer}}} \right)\end{matrix}}{\begin{matrix}{X_{{Cfe}\_ i} + {X_{{Cfe}\_ o}}} \\{\frac{X_{Cfluid}R_{fluid}}{X_{Cfluid} + R_{fluid}} + \frac{X_{Cpolymer}R_{polymer}}{X_{Cpolymer} + R_{polymer}}}\end{matrix}}}{{where}\text{:}}} & (1) \\{{X_{{Cfe}_{i}} = \frac{1}{{\mathbb{i}}\;\omega\; C_{{fe}\_ i}}},{X_{{Cfe}\_ o} = \frac{1}{{\mathbb{i}}\;\omega\; C_{{fe}\_ o}}},{X_{{Cpe}\_ o} = \frac{1}{{\mathbb{i}}\;\omega\; C_{{pe}\_ o}}},{X_{{Cpe}\_ i} = \frac{1}{{\mathbb{i}\omega}\; C_{{pe}\_ i}}},{X_{Cpolymer} = \frac{1}{{\mathbb{i}\omega}\; C_{polymer}}},{X_{Cfluid} = \frac{1}{{\mathbb{i}\omega}\; C_{fluid}}}} & (2)\end{matrix}$And the resistors are not frequency dependent (the voltage sourcefrequency is smaller than the minimum ionic rotation frequency withinthe fluid or polymer).

Capacitances and resistances are calculated with the coaxial model. As areminder:

-   -   the resistance measured between the inner electrode with radius        R_(i) and output electrode with radius R_(o) of a coaxial cable        of length l filled with a material of resistivity ρ is:

$\begin{matrix}{{R(l)} = \frac{{\rho ln}\frac{R_{o}}{R_{i}}}{2\pi\; l}} & (3)\end{matrix}$

-   -   the capacitance measured between the inner electrode with radius        R_(i) and output electrode with radius R_(o) of a coaxial cable        of length l filled with a material with a relative dielectric        constant ε_(dielectric) is:

$\begin{matrix}{{C(l)} = \frac{2{\pi ɛ}_{dielectric}ɛ_{0}l}{\ln\frac{R_{o}}{R_{i}}}} & (4)\end{matrix}$Interestingly enough, the time constant associated to this coax cable isindependent of any geometrical parameters of the cable:τ=R(l)C(l)=ε_(dielectric)ε₀ρ  (5)We will use the following notations:

$\begin{matrix}{{h_{electrodes} = h_{conductor}}{{ɛ_{dielectric}^{\prime} = {2{\pi ɛ}_{dielectric}ɛ_{0}}},{ɛ_{fluid}^{\prime} = {2{\pi ɛ}_{fluid}ɛ_{0}}},{ɛ_{polymer}^{\prime} = {2{\pi ɛ}_{polymer}ɛ_{0}}}}{{\rho_{fluid}^{\prime} = \frac{\rho_{fluid}}{2\pi}},{\rho_{polymer}^{\prime} = \frac{\rho_{polymer}}{2\pi}}}{{\tau_{polymer} = {{ɛ_{0}\rho_{polymer}ɛ_{polymer}} = {ɛ_{polymer}^{\prime}\rho_{polymer}^{\prime}}}},{\tau_{fluid} = {{ɛ_{0}\rho_{fluid}} = {ɛ_{fluid}^{\prime}\rho_{fluid}^{\prime}}}}}{\xi = \frac{h_{fluid}}{h_{electrodes}}}} & (6)\end{matrix}$The expressions for the capacitances instantiated by equations (2) are:

$\begin{matrix}{{C_{e\_ i} = {\frac{ɛ_{dielectric}^{\prime}}{\ln\left( {1 + \frac{t_{dielectric}}{R_{i}}} \right)}h_{electrodes}}}{C_{e\_ o} = {\frac{ɛ_{dielectric}^{\prime}}{\ln\left( {1 + \frac{t_{dielectric}}{R_{i} + t_{dielectric} + d_{electrodes}}} \right)}h_{electrodes}}}\begin{matrix}{{C_{{fe}\_ i}\left( h_{fluid} \right)} = {\frac{ɛ_{dielectric}^{\prime}}{\ln\left( {1 + \frac{t_{dielectric}}{R_{i}}} \right)}h_{fluid}}} \\{= {{\xi\;{C_{{fe}\_ i}\left( h_{electorodes} \right)}} = {\xi\; C_{e\_ i}}}}\end{matrix}\begin{matrix}{{C_{{fe}\_ o}\left( h_{fluid} \right)} = {\frac{ɛ_{dielectric}^{\prime}}{\ln\left( {1 + \frac{t_{dielectric}}{R_{i} + t_{dielectric} + d_{electrodes}}} \right)}h_{fluid}}} \\{= {\xi\;{C_{{fe}\_ o}\left( h_{electrodes} \right)}}} \\{= {\xi\; C_{e\_ o}}}\end{matrix}\begin{matrix}{{C_{{pe}\_ i}\left( h_{fluid} \right)} = {\frac{ɛ_{dielectric}^{\prime}}{\ln\left( {1 + \frac{t_{dielectric}}{R_{i}}} \right)}\left( {h_{electrodes} - h_{fluid}} \right)}} \\{= {\left( {1 - \xi} \right){C_{{pe}\_ i}\left( h_{electrodes} \right)}}} \\{= {\left( {1 - \xi} \right)C_{e\_ i}}}\end{matrix}\begin{matrix}{{C_{{pe}\_ i}\left( h_{fluid} \right)} = {\frac{ɛ_{dielectric}^{\prime}}{\ln\left( {1 + \frac{t_{dielectric}}{R_{i} + t_{dielectric} + d_{electrodes}}} \right)}\left( {h_{electrodes} - h_{fluid}} \right)}} \\{= {\left( {1 - \xi} \right){C_{{pe}\_ o}\left( h_{electrodes} \right)}}} \\{= {\left( {1 - \xi} \right)C_{e\_ o}}}\end{matrix}\begin{matrix}{{C_{fluid}\left( h_{fluid} \right)} = {\frac{ɛ_{fluid}^{\prime}}{\ln\left( {1 + \frac{d_{electrodes}}{R_{i} + t_{dielectric}}} \right)}h_{fluid}}} \\{= {\xi\;{C_{fluid}\left( h_{electrodes} \right)}}} \\{= {\xi\; C_{{FC},{fluid}}}}\end{matrix}\begin{matrix}{{C_{polymer}\left( h_{fluid} \right)} = {\frac{ɛ_{fluid}^{\prime}}{\ln\left( {1 + \frac{d_{electrodes}}{R_{i} + t_{dielectric}}} \right)}\left( {h_{electrodes} - h_{fluid}} \right)}} \\{= {\left( {1 - \xi} \right){C_{polymer}\left( h_{electrodes} \right)}}} \\{= {\left( {1 - \xi} \right)C_{{FPC},{polymer},0}}}\end{matrix}} & (7)\end{matrix}$

All capacitances are linear functions of h_(fluid)

The expressions for the resistances instantiated by equations (2) are:

$\begin{matrix}{\begin{matrix}{{R_{fluid}\left( h_{fluid} \right)} = \frac{\rho_{fluid}^{\prime}{\ln\left( {1 + \frac{d_{electrodes}}{R_{i} + t_{dielectric}}} \right)}}{h_{fluid}}} \\{= \frac{R_{fluid}\left( h_{electrodes} \right)}{\xi}} \\{= \frac{R_{{FC},{fluid}}}{\xi}}\end{matrix}\begin{matrix}{{R_{polymer}\left( h_{fluid} \right)} = \frac{\rho_{polymer}^{\prime}{\ln\left( {1 + \frac{d_{electrodes}}{R_{i} + t_{dielectric}}} \right)}}{h_{electrodes} - h_{fluid}}} \\{= \frac{R_{polymer}\left( h_{electrodes} \right)}{1 - \xi}} \\{= \frac{R_{{FPC},{polymer},0}}{1 - \xi}}\end{matrix}} & (8)\end{matrix}$

All conductances are linear functions of h_(fluid)

Step 1: Device in Air

FPC filled with polymer only

FC filled with air

Test cylinders filled with air.

Measuring the admittance {tilde over (Y)}_(C,test,air) of the testcylinders in the air (ε_(air)=1) allows the experimental determinationof the thickness of the dielectrics within the FPC and FC:

$\begin{matrix}\begin{matrix}{{\overset{\sim}{Y}}_{C,{test},{air}} = {{\mathbb{i}\omega}\; C_{test}}} \\{= \left. {{\mathbb{i}\omega}\frac{2{\pi ɛ}_{0}}{\ln\left( {1 + \frac{{2t_{dielectric}} + d_{electrodes}}{R_{i}}} \right)}h_{electrodes}}\Leftrightarrow t_{dielectric} \right.} \\{= {\frac{1}{2}\left( {{R_{i}\left( {{\mathbb{e}}^{\frac{2{\pi ɛ}_{0}{wh}_{electrodes}}{{??}{({\overset{\_}{Y}}_{C,{test},{air}})}}} - 1} \right)} - d_{electrodes}} \right)}}\end{matrix} & (9)\end{matrix}$

The admittance of the FC filled with air is (ρ_(air)≈∞, ε_(air)=1)

$\begin{matrix}\begin{matrix}{\frac{{\mathbb{i}\omega}\; h_{electrodes}}{{\overset{\sim}{Y}}_{C,{test},{air}}} = \left. \left( {{\frac{1}{ɛ_{dielectric}^{\prime}}{\ln\left( {1 + \frac{t_{dielectric}}{R_{i}}} \right)}} + {\frac{1}{ɛ_{air}^{\prime}}{\ln\left( {1 + \frac{d_{electrodes}}{R_{i} + t_{dielectric}}} \right)}} + {\frac{1}{ɛ_{dielectric}^{\prime}}{\ln\left( {1 + \frac{t_{dielectric}}{R_{i} + t_{dielectric} + d_{electrodes}}} \right)}}} \right)\Leftrightarrow \right.} \\{ɛ_{dielectric}^{\prime} = \frac{\ln\left( {\left( {1 + \frac{t_{dielectric}}{R_{i}}} \right)\left( {1 + \frac{t_{dielectric}}{R_{i} + t_{dielectric} + d_{electrodes}}} \right)} \right)}{\frac{\omega\; h_{electrodes}}{{??}\left( {\overset{\sim}{Y}}_{C,{test},{air}} \right)} - {\frac{1}{ɛ_{air}^{\prime}}{\ln\left( {1 + \frac{d_{electrodes}}{R_{i} + t_{dielectric}}} \right)}}}}\end{matrix} & (10)\end{matrix}$The measurements performed in air allowed the determinations of twocritical values: ε_(dielectric) and t_(dielectric)

Let us measure the FPC in air (h_(fluid)=0):

$\begin{matrix}{\mspace{79mu}\begin{matrix}{{\overset{\sim}{Z}}_{g,{FPC},{air}} = \left. {X_{{Cpe}\_ i} + X_{{Cpe}\_ o} + \frac{X_{Cpolymer}R_{polymer}}{X_{Cpolymer} + R_{polymer}}}\Leftrightarrow \right.} \\{{\overset{\sim}{Z}}_{g,{FPC},{air}} - \left( {X_{{Cpe}\_ i} + X_{{Cpe}\_ o}} \right)} \\{= \frac{R_{polymer}(0)}{1 + {{\mathbb{i}\omega}\;{R_{polymer}(0)}{C_{polymer}(0)}}}}\end{matrix}} & (11)\end{matrix}$

Replace the values for X_(Cpe) _(—) _(i) and X_(Cpe) _(—) _(o) with theexpression given in (7) for h_(fluid)=0:

$\begin{matrix}{{{\overset{\sim}{Z}}_{g,{FPC},{air}} - {\frac{1}{\mathbb{i}\omega}\left( {\frac{1}{C_{{pe}\_ i}(0)} + \frac{1}{C_{{pe}\_ o}}} \right)}} = {\left. \frac{R_{polymer}(0)}{1 + {{\mathbb{i}\omega}\;{R_{polymer}(0)}{C_{polymer}(0)}}}\Leftrightarrow{{\Re\left( {\overset{\sim}{Z}}_{g,{FPC},{air}} \right)} + {{\mathbb{i}??}\left( {\overset{\sim}{Z}}_{g,{FPC},{air}} \right)} + {\frac{\mathbb{i}}{\omega}\left( {\frac{1}{C_{{pe}\_ i}(0)} + \frac{1}{C_{{pe}\_ o}(0)}} \right)}} \right. = \left. \frac{\left( {1 - {\mathbb{i}\omega\tau}_{polymer}} \right){R_{polymer}(0)}}{1 + {\omega^{2}\tau_{polymer}^{2}}}\Leftrightarrow\left\{ {\begin{matrix}{{\Re\left( {\overset{\sim}{Z}}_{g,{FPC},{air}} \right)} = \frac{R_{polymer}(0)}{1 + {\omega^{2}\tau_{polymer}^{2}}}} \\{{{{??}\left( {\overset{\sim}{Z}}_{g,{FPC},{air}} \right)} + {\frac{1}{\omega}\left( {\frac{1}{C_{{pe}\_ i}(0)} + \frac{1}{C_{{pe}\_ o}(0)}} \right)}} = \left. \frac{{- {\omega\tau}_{polymer}}{R_{polymer}(0)}}{1 + {\omega^{2}\tau_{polymer}^{2}}}\Leftrightarrow \right.}\end{matrix}\left\{ \begin{matrix}{{\Re\left( {\overset{\sim}{Z}}_{g,{FPC},{air}} \right)} = \frac{R_{polymer}(0)}{1 + {\omega^{2}\tau_{polymer}^{2}}}} \\{{{{??}\left( {\overset{\sim}{Z}}_{g,{FPC},{air}} \right)} + {\frac{1}{\omega}\left( {\frac{1}{C_{{pe}\_ i}(0)} + \frac{1}{C_{{pe}\_ o}(0)}} \right)}} = \left. {{- {\omega\tau}_{polymer}}{\Re\left( {\overset{\sim}{Z}}_{g,{FPC},{air}} \right)}}\Leftrightarrow \right.}\end{matrix} \right.} \right. \right.}} & (12) \\{\mspace{79mu}\left\{ \begin{matrix}{\tau_{polymer} = \left. \frac{\frac{1}{C_{{pe}\_ i}(0)} + \frac{1}{C_{{pe}\_ o}(0)} + {{\omega??}\left( {\overset{\sim}{Z}}_{g,{FPC},{air}} \right)}}{\omega^{2}{\Re\left( Z_{g,{FPC},{air}} \right)}}\Leftrightarrow \right.} \\{{R_{polymer}(0)} = {\left( {1 + {\omega^{2}\tau_{polymer}^{2}}} \right){\Re\left( {\overset{\sim}{Z}}_{g,{FPC},{air}} \right)}}}\end{matrix} \right.} & \;\end{matrix}$Step 2: Measure the Fluid and Polymer Admittance at any Time after theDevice has been Immersed in a Fluid.

Assume the fluid etched the polymer, and got to the coordinate h_(fluid)of the FPC.

The FC complex admittance yields:

$\begin{matrix}{\mspace{79mu}\left\{ \begin{matrix}{\tau_{fluid} = \left. {- \frac{\frac{1}{C_{{pe}\_ i}(0)} + \frac{1}{C_{{pe}\_ o}(0)} + {{\omega??}\left( {\overset{\sim}{Z}}_{g,{FC}} \right)}}{\omega^{2}{\Re\left( Z_{g,{FC}} \right)}}}\Leftrightarrow \right.} \\{{R_{fluid}(0)} = {\left( {1 + {\omega^{2}\tau_{fluid}^{2}}} \right){\Re\left( {\overset{\sim}{Z}}_{g,{FC}} \right)}}}\end{matrix} \right.} & (13)\end{matrix}$

The FPC complex admittance yields:

$\begin{matrix}{{{{X\left( h_{fluid} \right)} + {X\left( h_{fluid} \right)} + \frac{{X_{Cfluid}\left( h_{fluid} \right)}{R_{fluid}\left( h_{fluid} \right)}}{{X_{Cfluid}(x)} + {R_{fluid}(x)}}} = {{\frac{1}{\xi}\left( {X_{C,{FC},{{fe}\_ i}} + X_{C,{FC},{{fe}\_ o}} + \frac{X_{C,{FC},{fluid}}R_{{FC},{fluid}}}{X_{C,{FC},{fluid}} + R_{{FC},{fluid}}}} \right)} = {\frac{1}{\xi}{\overset{\sim}{Z}}_{g,{FC}}}}}\mspace{79mu}{and}} & (14) \\{{{{{X_{{Cpe}\_ i}\left( h_{fluid} \right)} + {X_{{Cpe}\_ o}\left( h_{fluid} \right)} + \frac{{X_{Cfluid}\left( h_{fluid} \right)}{R_{fluid}\left( h_{fluid} \right)}}{{X_{Cfluid}(x)} + {R_{fluid}(x)}}} = {\frac{1}{1 - \xi}{\overset{\sim}{Z}}_{g,{FPC},0}}}\mspace{79mu}{{Then}\text{:}}}\mspace{56mu}} & (15) \\{\mspace{79mu}{{\overset{\sim}{Z}}_{g,{FC}} = {\left. \frac{\frac{1}{\xi}{\overset{\sim}{Z}}_{g,{FC}}\frac{1}{1 - \xi}{\overset{\sim}{Z}}_{g,{FPC},0}}{{\frac{1}{\xi}{\overset{\sim}{Z}}_{g,{FC}}} + {\frac{1}{1 - \xi}{\overset{\sim}{Z}}_{g,{FPC},0}}}\Leftrightarrow\mspace{79mu} h_{fluid} \right. = {h_{electrodes}\frac{{\overset{\sim}{Z}}_{g,{FC}}\left( {{\overset{\sim}{Z}}_{g,{FPC},0} - {\overset{\sim}{Z}}_{g,{FPC}}} \right)}{{\overset{\sim}{Z}}_{g,{FPC}}\left( {{\overset{\sim}{Z}}_{g,{FPC},0} - {\overset{\sim}{Z}}_{g,{FC}}} \right)}}}}} & (16)\end{matrix}$

With this complex impedance measurement approach, the etching of thepolymer with a FPC h_(fluid) is calculated based on the initial valuesof the impedance of the FPC (FPC in air) and the adjacent FPCmeasurement. This does not solve the variations of the polymerelectrical parameters vs. time.

Let us look at a frequency swipe method: for the same h_(fluid) thefrequency of the measurement of the AMA is changed within limits largerthan the poles and zeros of the complex admittance.

For an FC:

$\begin{matrix}{{\left. \left. \left. \mspace{79mu}\begin{matrix}{{\overset{\sim}{Z}}_{g,{FC}} = {{\frac{1}{\mathbb{i}\omega}\left( {\frac{1}{C_{{FC},{fe}_{i}}} + \frac{1}{C_{{FC},{fe}_{o}}}} \right)} + \frac{R_{{FC},{fluid}}}{1 + {\mathbb{i}\omega\tau}_{fluid}}}} \\{\left( {\frac{1}{C_{{FC},{fe}_{i}}} + \frac{1}{C_{{FC},{fe}_{o}}}} \right)\overset{\Delta}{=}\frac{1}{C_{{FC},{fe}}}}\end{matrix} \right\}\Rightarrow\begin{matrix}{{\overset{\sim}{Z}}_{g,{FC}} = {{\frac{1}{\mathbb{i}\omega}\frac{1}{C_{{FC},{fe}}}} + \frac{R_{{FC},{fluid}}}{1 + {\mathbb{i}\omega\tau}_{fluid}}}} \\{{R_{{FC},{fluid}}C_{{FC},{fe}}}\overset{\Delta}{=}\tau_{df}}\end{matrix} \right\}\Rightarrow{{\overset{\sim}{Z}}_{g,{FC}}} \right. = {\frac{1}{C_{{FC},{fe}}}\frac{\sqrt{{\omega^{2}\tau_{fd}^{2}} + \left( {1 + {\omega^{2}\left( {\tau_{fluid}^{2} + {\tau_{fluid}\tau_{df}}} \right)}} \right)^{2}}}{\omega\left( {1 + {\omega^{2}\tau_{fluid}^{2}}} \right)}}}{{The}\mspace{14mu}{measured}\mspace{14mu}{absolute}\mspace{14mu}{value}\mspace{14mu}{of}\mspace{14mu}{the}\mspace{14mu}{impedance}\mspace{14mu}{of}\mspace{14mu}{the}\mspace{14mu}{FC}\mspace{14mu}{has}\mspace{14mu}{the}\mspace{14mu}{poles}\mspace{14mu}{and}\mspace{14mu}{zeros}\mspace{14mu}{placed}\mspace{14mu}{at}\text{:}}} & (17) \\{{{{\omega_{pole} \in \left\{ {0,{- \frac{i}{\tau_{fluid}}},\frac{i}{\tau_{fluid}}} \right\}},{\omega_{zero} \in \left\{ {{- \frac{i}{\tau_{fluid} + \tau_{df}}},\frac{i}{\tau_{fluid} + \tau_{df}}} \right\}}}\mspace{85mu}{{with}\text{:}}\mspace{79mu}{{R_{{FC},{fluid}}C_{{FC},{fe}}}\overset{\Delta}{=}\tau_{df}}}{\tau_{df} = {ɛ_{dielectric}^{\prime}\rho_{fluid}^{\prime}\frac{\ln\left( {1 + \frac{d_{electrodes}}{R_{i} + t_{dielectric}}} \right)}{\begin{matrix}{{\ln\left( {1 + \frac{t_{dielectric}}{R_{i}}} \right)}{\ln\left( {1 + \frac{t_{dielectric}}{R_{i} + t_{dielectric} + d_{electrodes}}} \right)}} \\{\ln\left( {\left( {1 + \frac{t_{dielectric}}{R_{i} + t_{dielectric} + d_{electrodes}}} \right)\left( {1 + \frac{t_{dielectric}}{R_{i}}} \right)} \right)}\end{matrix}}}}} & (18)\end{matrix}$

From the bode plot of the absolute value of the complex impedance of anFC, one can extract τ_(fluid)+τ_(df) and τ_(fluid) which is equivalentto resolving ρ_(fluid) and ε_(fluid).

We will use the same procedure for extracting the parameters of thepolymer within an FPC:

$\begin{matrix}{{{\overset{\sim}{Y}}_{g,{FPC}} = {\left. {\frac{1}{Z_{g,{FPC},{polymer}}} + \frac{1}{Z_{g,{FPC},{fluid}}}}\Leftrightarrow{\overset{\sim}{Y}}_{g,{FPC}} \right. = {\left. {\frac{1}{\frac{1}{\xi}{\overset{\sim}{Z}}_{g,{FC}}} + \frac{1}{\frac{1}{1 - \xi}{\overset{\sim}{Z}}_{g,{FPC},{fluid}}}}\Leftrightarrow Y_{g,{FPC}} \right. = \frac{C_{{FC},{ee}}\omega}{\begin{matrix}\left\lbrack {{\omega\tau}_{df} - {{\mathbb{i}}\left( {1 + {\omega^{2}{\tau_{fluid}\left( {\tau_{fluid} + \tau_{df}} \right)}}} \right)}} \right\rbrack \\\left\lbrack {{\omega\tau}_{dp} - {{\mathbb{i}}\left( {1 + {\omega^{2}{\tau_{polymer}\left( {\tau_{polymer} + \tau_{dp}} \right)}}} \right)}} \right\rbrack\end{matrix}}}}}\begin{Bmatrix}{{{\xi\left\lbrack {{\omega\tau}_{dp} - {{\mathbb{i}}\left( {1 + {\omega^{2}{\tau_{polymer}\left( {\tau_{polymer} + \tau_{dp}} \right)}}} \right)}} \right\rbrack}\left( {1 + {\omega^{2}\tau_{fluid}^{2}}} \right)} +} \\{{\left( {1 - \xi} \right)\left\lbrack {{\omega\tau}_{df} - {{\mathbb{i}}\left( {1 + {\omega^{2}{\tau_{fluid}\left( {\tau_{fluid} + \tau_{df}} \right)}}} \right)}} \right\rbrack}\left( {1 + {\omega^{2}\tau_{polymer}^{2}}} \right)}\end{Bmatrix}} & (19)\end{matrix}$The poles and zeros for {tilde over (Y)}_(g,FPC) can be estimated fromthe Bode plot ∥{tilde over (Y)}_(g,FPC)∥, therefore one can extractρ_(polymer) and ε_(polymer) at any time during the measurement of thefluid.

Determination of Chemical Active Species Concentration

Let us consider an Admittance Measurement Array (AMA) as describedabove, with the following characteristics:

-   -   Confines a (not necessarily) square array of        N_(cylinders)×N_(cylinders) cylinders.    -   4 cylinders of the AMA are not covered with Si₃N₄/SiO₂        dielectrics and are used to measure (in air) the specifics of        the dielectric covering the A1 electrodes (ε_(dielectric)        t_(dielectric)).    -   N_(cylinder)=N_(FC,cylinder) cylinders (different from the 4        above) are not filled with any polymers.    -   N_(polymer)=N_(cylinders)×N_(cylinders)−N_(FC,cylinder)−4 are        filled with N_(polymer) different types of polymers.    -   Every polymer is chemically sensitive to a specific chemical        species within the fluid under test. For example, assume the        fluid under test confines N_(specie) active species with        concentrations C_(specie). The polymer in cylinder k ∈≦{1, . . .        , N_(polymer)}⊂N reacts with the active species within the        fluid. As a result of the reactions, the polymer will become        soluble within the solution. The rate of solubility is related        to the etching rate of the polymer with a function isomorphic        with:        r _(ks) =m _(ks)(C _(s) −C _(ks,threshold))θ(C _(s) −C        _(ks,threshold)),∀k={1, . . . ,N _(polymer) },∀s={1, . . . ,N        _(species)}  (20)        where:    -   r_(ks) Etching rate of polymer k reacting with specie s    -   θ(C_(s)−C_(ks,threshold)) Heaviside function of difference        between active specie concentration Cs and concentration        threshold Cks,threshold above which polymer k reacts with specie        s    -   M_(ks)(C_(s)−C_(ks,threshold)) Increasingly monotonic function        describing the etching rate of polymer k by specie s of        concentration C_(s).

The total etch rate for polymer k while reacting to all speciesN_(specie) within the fluid is the sum of the etch rate of the polymerper specie:

$\begin{matrix}{\mspace{79mu}{{r_{k} = {{\sum\limits_{i = 1}^{N_{species}}\; r_{ks}} = {\sum\limits_{i = 1}^{N_{species}}\;{M_{ki}\left( {C_{i} - C_{{ki},{threshold}}} \right)}}}},\mspace{79mu}{{\forall k} = \left\{ {1,\ldots\mspace{14mu},N_{polymer}} \right\}},{{\forall i} = \left\{ {1,\ldots\mspace{14mu},N_{species}} \right\}}}} & (21)\end{matrix}$withM_(id)(C_(i)−C_(id,threshold))=m_(ks)(C_(s)−C_(ks,threshold))θ(C_(s)−C_(ks,threshold)).The AMA structure is measured in air and the specifics of all Npolymerare stored as (ε_(k), ρ_(k), τ_(k)), ∀k={1, . . . , N_(polymer)}.

The AMA structure immersed in the solution under test. We will assume atany time there is no concentration gradient of any active species at thesurface of the AMA, therefore all cylinders “see” at the same time thesame value of the concentration value for any species.

The admittance measurement circuit measures for every cylinder (FluidPolymer filled Cylinders (FPF) and Fluid-filled Cylinders (FC)) at asample rate S all complex admittances and calculates:

For every FC

$\mspace{79mu}{\left( {ɛ_{{FC},{fluid},e^{\prime}}^{(t_{j})},\rho_{{FC},{fluid},e^{\prime}}^{(t_{j})},\tau_{{FC},{fluid},e}^{(t_{j})}} \right),{{\forall c} = \left\{ {1,\ldots\mspace{14mu},N_{{FC},{cylinder}}} \right\}},\mspace{79mu}{{\forall t_{j}} = \left\{ {\frac{1}{S},\ldots\mspace{14mu},\frac{N}{S}} \right\}},{N \in N}}$

The average values of (ε_(FC,fluid) ^((t) ^(j) ⁾,ρ_(FC,fluid) ^((t) ^(j)⁾,τ_(FC,fluid) ^((t) ^(j) ⁾) at time t_(j):

$\begin{matrix}{\mspace{79mu}{{ɛ_{{FC},{fluid}}^{(t_{j})} = {\frac{1}{N_{{FC},{cylinder}}}{\sum\limits_{i = 1}^{N_{{FC},{cylinder}}}ɛ_{{FC},{fluid},i}^{(t_{j})}}}}\;\mspace{20mu}{\rho_{{FC},{fluid}}^{(t_{j})} = {\frac{1}{N_{{FC},{cylinder}}}{\sum\limits_{i = 1}^{N_{{FC},{cylinder}}}\;\rho_{{FC},{fluid},i}^{(t_{j})}}}}\mspace{20mu}{\tau_{{FC},{fluid}}^{(t_{j})} = {\frac{1}{N_{{FC},{cylinder}}}{\sum\limits_{i = 1}^{N_{{FC},{cylinder}}}\;\tau_{{FC},{fluid},i}^{(t_{j})}}}}}} & (22)\end{matrix}$

-   -   Based upon the values obtained above, assuming the fluid        parameters within adjacent FC and PFC are the same (no        concentration gradient at the surface of the AMA), calculate for        every FPC

$\mspace{20mu}{\left( {ɛ_{{polymer},k}^{(t_{j})},\rho_{{polymer},k}^{(t_{j})},\tau_{{polymer},k}^{(t_{j})},h_{{fluid},k}^{(t_{j})}} \right),{{\forall k} = \left\{ {1,\ldots\mspace{14mu},N_{polymer}} \right\}},\mspace{20mu}{{\forall t_{j}} = \left\{ {\frac{1}{S},\ldots\mspace{14mu},\frac{N}{S}} \right\}},{N \in {{\mathbb{N}}.}}}$

We can approximate the etching rate at time t_(j) as:

$\begin{matrix}{\mspace{79mu}{{{{\overset{\sim}{r}}_{k}^{(t_{j})} \approx \frac{h_{{fluid},k}^{(t_{j})} - h_{{fluid},k}^{(t_{j - 1})}}{t_{j} - t_{j - 1}}} = {\left( {h_{{fluid},k}^{(t_{j})} - h_{{fluid},k}^{(t_{j - 1})}} \right)S}},\mspace{20mu}{{\forall k} = \left\{ {1,\ldots\mspace{14mu},N_{polymer}} \right\}}}} & (23)\end{matrix}$

Therefore for any time t_(j) we get:

$\begin{matrix}{\mspace{79mu}{{{\overset{\sim}{r}}_{k}^{(t_{j})} = {\sum\limits_{i = 1}^{N_{species}}\;{M_{ki}\left( {C_{i} - C_{{ki},{threshold}}} \right)}}},\mspace{79mu}{{\forall k} = \left\{ {1,\ldots\mspace{14mu},N_{polymer}} \right\}},{{\forall i} = \left\{ {1,\ldots\mspace{14mu},N_{species}} \right\}}}} & (24)\end{matrix}$

We can express the relation above in matrix form:

{tilde over (r)}_([N) _(polymer) _(,1]) ^((t) ^(j) ⁾=

{tilde over (r)}_(k) ^((t) ^(j) ⁾

, measured etching rate matrix, at time t_(j)

{tilde over (C)}_([1,N) _(specie) _(]) ^((t) ^(j) ⁾=

{tilde over (c)}_(i) ^((t) ^(j) ⁾

^(, concentration matrix at time t) _(j), to be determined

C_(TH[N) _(polymer) _(,N) _(specie) _(])=

C_(ki,threshold)

, concentrating threshold for polymer k reacting with specie I, known(25)

M_([N) _(polymer) _(,N) _(specie) _(])(C)=

M_(ik)(C_(ik))

, etching rate function matrix for polymer k, reacting with specie i,known

C_(H[N) _(polymer) _(,N) _(specie) _(]) ^()t) ^(j) ⁾=1_([N) _(polymer)_(,t]){tilde over (C)}_([1,N) _(specie) _(]) ^((t) ^(j) ⁾−C_(TH[N)_(polymer) _(,N) _(specie) _(]), normalized concentration matrix

The systems of equations from (24) can be written as:{tilde over (r)} _([N) _(polymer) _(,t]) ^((t) ^(j) ⁾ =M _([N)_(polymer) _(,N) _(specie) _(])(C _(H[N) _(polymer) _(,N) _(specie) _(])^((t) ^(j) ⁾)1_([N) _(polymer) _(,1])  (26)

For the case of when the matrix M_([N) _(polymer) _(,N) _(specie) _(])is has an inverse, i.e., N_(polymer)=N_(species)=N_(e), the equationreads:{tilde over (r)} _([N) _(e) _(t]) ^((t) ^(j) ⁾ =M _([N) _(e) _(,N) _(e)_(])(C _(H[N) _(e) _(,N) _(e) _(]) ^((t) ^(j) ⁾)1_([N) _(e) _(,1])  (27)

And after some linear transformations becomes:M _([N) _(e) _(,N) _(e) _(])(1_([N) _(e) _(,1]) {tilde over (c)} _([1,N)_(e) _(]) ^((t) ^(j) ⁾ −C _(TH[N) _(e) _(,N) _(e) _(])){tilde over (r)}_([N) _(e) _(,1]) ^((t) ^(j) ⁾=1_([N) _(e) _(,1])  (28)

This represents a system of N_(e) equations N_(e) unknowns {tilde over(C)}_([1,N) _(e) _(]) ^((t) ^(j) ⁾, which all have a unique solution, aslong as the matrix M_([N) _(e) _(,N) _(e) _(]) is inverseable.

Although the invention is illustrated and described herein withreference to specific embodiments, the invention is not intended to belimited to the details shown. Rather, various modifications may be madein the details within the scope and range of equivalents of the claimsand without departing from the invention.

The invention claimed is:
 1. Apparatus for electrochemical analysis offluids, comprising: a chamber having a depth dimension for accommodatinga volume of a fluid under test; a first electrode disposed within saidchamber and extending therewithin along said depth dimension; a secondelectrode disposed within said chamber and extending therewithin alongsaid depth dimension in laterally spaced relation with said firstelectrode; and a soluble solid disposed within said chamber between saidfirst and second electrodes so as to substantially completely occupy alateral gap therebetween to an extent of at least a portion of saiddepth dimension; wherein a rate of dissolution of said soluble solidwithin said fluid is at least partially dependent on a chemicalconcentration of a corresponding analyte present in solution in saidfluid.
 2. An apparatus according to claim 1, wherein said soluble solidis an annealed polymer.
 3. An apparatus according to claim 1, whereinsaid soluble solid substantially completely occupies said lateral gapbetween said first and second electrodes to a substantially full extentof a depthwise overlap between said first and second electrodes withinsaid chamber.
 4. An apparatus according to claim 3, wherein said solublesolid substantially completely fills said chamber from an uppermostextent thereof downward through said substantially full extent of saiddepthwise overlap.
 5. An apparatus according to claim 1, wherein each ofsaid first and second electrodes at least partially defines an interiorwall of said chamber extending downward therewithin to an extent of adepthwise overlap between said first and second electrodes within saidchamber.
 6. An apparatus according to claim 5, wherein said firstelectrode is connected to ground and defines a substantially straightand cylindrically shaped wall extending downward within said chamber tosaid extent of said depthwise overlap, and said second electrode isconnected to a power supply and defines a substantially straight andcolumn shaped lateral wall extending substantially coaxially with saidcylindrical lateral wall, and downward within said chamber to saidextent of said depthwise overlap.
 7. An apparatus according to claim 1,wherein said apparatus comprises a measurement tool including asilicon-based integrated circuit device defining an upper margin; and anarray of grouped instances of said chamber, said first and secondelectrodes, and said soluble solid disposed along said upper margin topermit each of said instances of said soluble solid to be directlyexposed to said fluid under test; said integrated circuit device furtherincluding an electrode selector unit for permitting said integratedcircuit device to selectably individually apply a predetermined testsignal to each paired instance of said first and second electrode, andto receive in response thereto an analog signal corresponding to aprevailing electrical condition within the corresponding instance ofsaid chamber for use in determining said chemical concentration of saidcorresponding analyte present in solution in said fluid.
 8. An apparatusaccording to claim 7, wherein said prevailing electrical condition isselected from the group comprising conductance, complex admittance, andcomplex impedance.
 9. An apparatus according to claim 7, wherein saidintegrated circuit device further includes a measurement unit fordetermining a corresponding value of said prevailing electricalcondition based on said analog signal response.
 10. An apparatusaccording to claim 9, wherein said integrated circuit device furtherincludes a data processor unit for controlling said electrode selectorunit, receiving an analog signal from said measurement unitcorresponding to said determined value of said prevailing electricalcondition, and determining, based on said determined value of saidprevailing electrical condition, a chemical concentration of saidcorresponding analyte present in solution in said fluid.
 11. Anapparatus according to claim 10, wherein said apparatus is adapted todetermine said chemical concentration of said corresponding analytepresent in solution in said fluid without absolute calibration.
 12. Anapparatus according to claim 10, wherein said integrated circuit devicefurther includes an input/output data block for interfacing saidintegrated circuit device with at least one external device, includingreceiving and passing on to said at least one external device a digitalsignal from said data processor unit corresponding to a value of saidchemical concentration.
 13. An apparatus according to claim 7, whereinsaid integrated circuit further includes a control unit for controllingsaid test signal with respect to at least one of frequency andamplitude.
 14. An apparatus according to claim 7, wherein with respectto said array of grouped instances, said instances of said soluble solidexhibit a plurality of variations of said soluble solid, each saidvariation being associated with a rate of dissolution within said fluidat least partially dependent on a respectively different chemicalconcentration of said corresponding analyte present in solution in saidfluid.
 15. An apparatus according to claim 7, wherein with respect tosaid array of grouped instances, said instances of said soluble solidexhibit a plurality of variations of said soluble solid, each saidvariation being associated with a rate of dissolution within said fluidat least partially dependent on a chemical concentration of arespectively different corresponding analyte present in solution in saidfluid.
 16. An apparatus according to claim 7, wherein upon acommencement of a dissolution of said soluble solid into said fluid,said apparatus remains operable for purposes of measuring aconcentration of said analyte present in solution in said fluid at leastuntil said soluble solid no longer substantially completely occupiessaid lateral gap between said first and second electrodes.
 17. Anapparatus according to claim 7, wherein said integrated circuit deviceincludes a CMOS die, and said array of grouped instances is formed alongsaid upper margin via an associated appropriate MEMS process.
 18. Anapparatus according to claim 17, wherein said MEMS process includesforming a metallic contact pattern along said upper margin for providingconnectivity between said integrated circuit device and said array ofgrouped instances, and forming said paired first and second electrodesof each of said grouped instances via at least one instance of metallicdeposition followed by an associated appropriate etch.
 19. An apparatusaccording to claim 18, wherein forming said paired first and secondelectrodes of each instance of said grouped instances further includesat least one instance of depositing a material layer selected from thegroup consisting of SiO₂ and Si₃N₄, followed by an associatedappropriate etch, wherein at least one instance of a chamber of saidgrouped instances is suitably masked to prevent said deposition of saidmaterial layer, and is further kept substantially free of any of saidsoluble solid, so as to permit said chambers of said at least oneinstance of a chamber to be utilized for control purposes duringassociated analyte concentration measurement sessions.
 20. An apparatusaccording to claim 7, wherein at least one instance of a chamber of saidgrouped instances is kept substantially free of any of said solublesolid, so as to permit each chamber of said at least one instance of achamber to be substantially entirely filled with said fluid under testand thereby utilized as a reference chamber for monitoring changes inelectrical characteristics in said fluid under test during associatedanalyte concentration measurement sessions.